Absolute Value: Does
|x|^2 = x^2?
Proof that |x|^2 = x^2, assuming x is a real number.
Note that |x| = x/sgn(x), where sgn(x) is the sign function where:
sgn(x) = -1 if x < 0,
sgn(x) = 0 if x = 0,
and sgn(x) = 1 if x > 0
Case: x = 0.
Then:
|0| = 0 and |0|^2 = 0^2 = 0.
Case: x ≠ 0.
Then:
|x|^2 = (x/sgn(x))^2
= x/sgn(x) * x/sgn(x)
= x^2/sgn(x)^2
If x < 0, sgn(x) = -1, and since -1 * -1 = 1, sgn(x)^2 =
1
If x > 0, sgn(x) = 1, and since 1 * 1 = 1, sgn(x)^2 = 1
Hence:
x^2/sgn(x)^2
= x^2
QED
Caution: The statement
|x|^2 = x^2 is not true for complex numbers where the imaginary part is
nonzero.
Let x = a + b*i
|x|^2 = |a + b*i|^2 = (√(a^2 + b^2))^2 = a^2 + b^2
x^2 = (a + b*i)^2 = a^2 + 2*a*b*i – b^2 ≠ a^2 + b^2 (b ≠ 0)
Conclude: |x|^2 = x^2
only if x is a real number.
Eddie
This blog is property of Edward Shore, 2016
